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For example, if $X = \\cap_{i=1}^r D_i \\subset G/P$ is a general complete intersection of $r$ ample divisors such that $K_{G/P}^* \\otimes \\mathcal{O}_{G/P}(-\\sum_i D_i)$ is ample, then $X$ is Fano. We first classify these Fano complete intersections which are locally rigid. It turns out that most of them are hyperplane sections. 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